JEE Advanced 2007 Paper 1Question 1
Let the vectors $\overrightarrow{PQ}, \overrightarrow{QR}, \overrightarrow{RS}, \overrightarrow{ST}, \overrightarrow{TU}$ and $\overrightarrow{UP}$ represent the sides of a regular hexagon.
STATEMENT-1 : $\overrightarrow{PQ}\times\left(\overrightarrow{RS}+\overrightarrow{ST}\right)\neq\vec{0}$.
because
STATEMENT-2 : $\overrightarrow{PQ}\times\overrightarrow{RS}=\vec{0}$ and $\overrightarrow{PQ}\times\overrightarrow{ST}\neq\vec{0}$.
Vector Algebra
Solution
In a regular hexagon $\overrightarrow{ST}\parallel\overrightarrow{PQ}$ (opposite sides) but $\overrightarrow{RS}$ is not parallel to $\overrightarrow{PQ}$. So $\overrightarrow{PQ}\times\overrightarrow{RS}\ne\vec0$ and $\overrightarrow{PQ}\times\overrightarrow{ST}=\vec0$; Statement-2 is false, while $\overrightarrow{PQ}\times(\overrightarrow{RS}+\overrightarrow{ST})=\overrightarrow{PQ}\times\overrightarrow{RS}\ne\vec0$, so Statement-1 is true.